Question

If $f(x) = \left| \begin{matrix} 1 & x & (x + 1) \\ 2x & x(x - 1) & (x + 1)x \\ 3x(x - 1) & x(x - 1)(x - 2) & (x + 1)x(x - 1) \end{matrix} \right|$, then

$f(100)$ is equal to

• A. 0
• B. 1
• C. 100
• D. –100

Answer 1

Answer: (A)

0

Answer 2

1 & x & (x + 1) \\ 2x & x(x - 1) & (x + 1)x \\ 3x(x - 1) & x(x - 1)(x - 2) & (x + 1)x(x - 1) \end{matrix} \right|$$ Applying $C_{3} \rightarrow C_{3} - C_{2}$, we get $f(x) = \left| \begin{matrix} 1 & x & 1 \\ 2x & x(x - 1) & 2x \\ 3x(x - 1) & x(x - 1)(x - 2) & 3x(x - 1) \end{matrix} \right| = 0$. Hence $$f(100) = 0$$